49 ideas
| 162 | Can we understand an individual soul without knowing the soul in general? [Plato] |
| Full Idea: Do you think it possible to form an adequate conception of the nature of an individual soul without considering the nature of soul in general? | |
| From: Plato (Phaedrus [c.366 BCE], 270c) | |
| A reaction: Do animals understand anything (as opposed to simply being aware of things)? |
| 160 | The highest ability in man is the ability to discuss unity and plurality in the nature of things [Plato] |
| Full Idea: When I believe that I have found in anyone the ability to discuss unity and plurality as they exist in the nature of things, I follow his footsteps as if he was a god. | |
| From: Plato (Phaedrus [c.366 BCE], 266b) | |
| A reaction: This sounds like the problem of identity, which is at the heart of modern metaphysics. |
| 166 | A speaker should be able to divide a subject, right down to the limits of divisibility [Plato] |
| Full Idea: A speaker must be able to define a subject generically, and then to divide it into its various specific kinds until he reaches the limits of divisibility. | |
| From: Plato (Phaedrus [c.366 BCE], 277b) |
| 10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
| Full Idea: While truth can be defined in a relative way, as truth in one particular model, a non-relative notion of truth is implied, as truth in all models. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: [The article is actually discussing arithmetic] This idea strikes me as extremely important. True-in-all-models is usually taken to be tautological, but it does seem to give a more universal notion of truth. See semantic truth, Tarski, Davidson etc etc. |
| 10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
| Full Idea: In standard ZFC ('Zermelo-Fraenkel with Choice') set theory we deal merely with pure sets, not with additional urelements. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: The 'urelements' would the actual objects that are members of the sets, be they physical or abstract. This idea is crucial to understanding philosophy of mathematics, and especially logicism. Must the sets exist, just as the urelements do? |
| 10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
| Full Idea: In second-order logic there are three kinds of variables, for objects, for functions, and for predicates or sets. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: It is interesting that a predicate seems to be the same as a set, which begs rather a lot of questions. For those who dislike second-order logic, there seems nothing instrinsically wicked in having variables ranging over innumerable multi-order types. |
| 10165 | 'Analysis' is the theory of the real numbers [Reck/Price] |
| Full Idea: 'Analysis' is the theory of the real numbers. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: 'Analysis' began with the infinitesimal calculus, which later built on the concept of 'limit'. A continuum of numbers seems to be required to make that work. |
| 10174 | Mereological arithmetic needs infinite objects, and function definitions [Reck/Price] |
| Full Idea: The difficulties for a nominalistic mereological approach to arithmetic is that an infinity of physical objects are needed (space-time points? strokes?), and it must define functions, such as 'successor'. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: Many ontologically austere accounts of arithmetic are faced with the problem of infinity. The obvious non-platonist response seems to be a modal or if-then approach. To postulate infinite abstract or physical entities so that we can add 3 and 2 is mad. |
| 10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price] |
| Full Idea: A common formulation of Peano Arithmetic uses 2nd-order logic, the constant '1', and a one-place function 's' ('successor'). Three axioms then give '1 is not a successor', 'different numbers have different successors', and induction. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: This is 'second-order' Peano Arithmetic, though it is at least as common to formulate in first-order terms (only quantifying over objects, not over properties - as is done here in the induction axiom). I like the use of '1' as basic instead of '0'! |
| 10172 | Set-theory gives a unified and an explicit basis for mathematics [Reck/Price] |
| Full Idea: The merits of basing an account of mathematics on set theory are that it allows for a comprehensive unified treatment of many otherwise separate branches of mathematics, and that all assumption, including existence, are explicit in the axioms. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: I am forming the impression that set-theory provides one rather good model (maybe the best available) for mathematics, but that doesn't mean that mathematics is set-theory. The best map of a landscape isn't a landscape. |
| 10167 | Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price] |
| Full Idea: Structuralism has emerged from the development of abstract algebra (such as group theory), the creation of axiom systems, the introduction of set theory, and Bourbaki's encyclopaedic survey of set theoretic structures. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: In other words, mathematics has gradually risen from one level of abstraction to the next, so that mathematical entities like points and numbers receive less and less attention, with relationships becoming more prominent. |
| 10169 | Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price] |
| Full Idea: Relativist Structuralism simply picks one particular model of axiomatised arithmetic (i.e. one particular interpretation that satisfies the axioms), and then stipulates what the elements, functions and quantifiers refer to. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: The point is that a successful model can be offered, and it doesn't matter which one, like having any sort of aeroplane, as long as it flies. I don't find this approach congenial, though having a model is good. What is the essence of flight? |
| 10179 | There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price] |
| Full Idea: The term 'structure' has two uses in the literature, what can be called 'particular structures' (which are particular relational systems), but also what can be called 'universal structures' - what particular systems share, or what they instantiate. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §6) | |
| A reaction: This is a very helpful distinction, because it clarifies why (rather to my surprise) some structuralists turn out to be platonists in a new guise. Personal my interest in structuralism has been anti-platonist from the start. |
| 10181 | Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price] |
| Full Idea: According to 'pattern' structuralism, what we study are not the various particular isomorphic models of arithmetic, but something in addition to them: a corresponding pattern. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §7) | |
| A reaction: Put like that, we have to feel a temptation to wield Ockham's Razor. It's bad enough trying to give the structure of all the isomorphic models, without seeking an even more abstract account of underlying patterns. But patterns connect to minds.. |
| 10182 | There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price] |
| Full Idea: There are four main variants of structuralism in the philosophy of mathematics - formalist structuralism, relativist structuralism, universalist structuralism (with modal variants), and pattern structuralism. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §9) | |
| A reaction: I'm not sure where Chihara's later book fits into this, though it is at the nominalist end of the spectrum. Shapiro and Resnik do patterns (the latter more loosely); Hellman does modal universalism; Quine does the relativist version. Dedekind? |
| 10168 | Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price] |
| Full Idea: Formalist Structuralism endorses structural methodology in mathematics, but rejects semantic and metaphysical problems as either meaningless, or purely formal, or as inference relations. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §3) | |
| A reaction: [very compressed] I find the third option fairly congenial, certainly in preference to rather platonist accounts of structuralism. One still needs to distinguish the mathematical from the non-mathematical in the inference relations. |
| 10178 | Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price] |
| Full Idea: It is tempting to take a modal turn, and quantify over all possible objects, because if there are only a finite number of actual objects, then there are no models (of the right sort) for Peano Arithmetic, and arithmetic is vacuously true. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: [compressed; Geoffrey Hellman is the chief champion of this view] The article asks whether we are not still left with the puzzle of whether infinitely many objects are possible, instead of existent. |
| 10176 | Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price] |
| Full Idea: Universalist Structuralism is a semantic thesis, that an arithmetical statement asserts a universal if-then statement. We build an if-then statement (using quantifiers) into the structure, and we generalise away from any one particular model. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: There remains the question of what is distinctively mathematical about the highly generalised network of inferences that is being described. Presumable the axioms capture that, but why those particular axioms? Russell is cited as an originator. |
| 10177 | Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price] |
| Full Idea: Universalist Structuralism is eliminativist about abstract objects, in a distinctive form. Instead of treating the base element (say '1') as an ambiguous referring expression (the Relativist approach), it is a variable which is quantified out. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: I am a temperamental eliminativist on this front (and most others) so this is tempting. I am also in love with the concept of a 'variable', which I take to be utterly fundamental to all conceptual thought, even in animals, and not just a trick of algebra. |
| 10171 | The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price] |
| Full Idea: Relativist Structuralism must first assume the existence of an infinite set, otherwise there would be no model to pick, and arithmetical terms would have no reference. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: See Idea 10169 for Relativist Structuralism. They point out that ZFC has an Axiom of Infinity. |
| 7953 | Reasoning needs to cut nature accurately at the joints [Plato] |
| Full Idea: In our reasoning we need a clear view of the ability to divide a genus into species, observing the natural joints, not mangling any of the parts, like an unskilful butcher. | |
| From: Plato (Phaedrus [c.366 BCE], 265d) | |
| A reaction: In modern times this Platonic idea has become the standard metaphor for realism. I endorse it. I think nature has joints, and we should hunt for them. There are natural sets. The joints may exist in abstract concepts, as well as in objects. |
| 16121 | I revere anyone who can discern a single thing that encompasses many things [Plato] |
| Full Idea: If I believe that someone is capable of discerning a single thing that is also by nature capable of encompassing many, I follow 'straight behind, in his footsteps, as if he were a god'. | |
| From: Plato (Phaedrus [c.366 BCE], 266b) | |
| A reaction: [Plato quote Odyssey 2.406] This is the sort of simple but profound general observation which only the early philosophers bothered to make, and no one comments on now. Encompassing many under one is the very essence of thinking. |
| 153 | It takes a person to understand, by using universals, and by using reason to create a unity out of sense-impressions [Plato] |
| Full Idea: It takes a man to understand by the use of universals, and to collect out of the multiplicity of sense-impressions a unity arrived at by a process of reason. | |
| From: Plato (Phaedrus [c.366 BCE], 249b) |
| 154 | We would have an overpowering love of knowledge if we had a pure idea of it - as with the other Forms [Plato] |
| Full Idea: What overpowering love knowledge would inspire if it could bring a clear image of itself before our sight, and the same may be said of the other forms. | |
| From: Plato (Phaedrus [c.366 BCE], 250d) | |
| A reaction: the motivation in Plato's theory |
| 10173 | A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price] |
| Full Idea: One way for a nominalist to reject appeal to all abstract objects, including sets, is to only appeal to nominalistically acceptable objects, including mereological sums. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: I'm suddenly thinking that this looks very interesting and might be the way to go. The issue seems to be whether mereological sums should be seen as constrained by nature, or whether they are unrestricted. See Mereology in Ontology...|Intrinsic Identity. |
| 151 | True knowledge is of the reality behind sense experience [Plato] |
| Full Idea: True knowledge is concerned with the abode of true reality, without colour or shape, intangible but utterly real, apprehensible only to the intellect. | |
| From: Plato (Phaedrus [c.366 BCE], 247c) |
| 165 | If the apparent facts strongly conflict with probability, it is in everyone's interests to suppress the facts [Plato] |
| Full Idea: There are some occasions when both prosecution and defence should positively suppress the facts in favour of probability, if the facts are improbable. | |
| From: Plato (Phaedrus [c.366 BCE], 272e) |
| 9296 | The soul is self-motion [Plato] |
| Full Idea: Self-motion is of the very nature of the soul. | |
| From: Plato (Phaedrus [c.366 BCE], 245e) | |
| A reaction: This culminates a length discussion of the soul. He gives an implausible argument that the soul is immortal, because it could never cease its self-motion. Why are we so unimpressed by motion, when the Greeks were amazed by it? |
| 3654 | The pineal gland links soul to body, and unites the two symmetrical sides of the body [Descartes, by PG] |
| Full Idea: The soul is united with the body in just one place, a gland (the pineal) in the centre of the brain. It is placed so that its slightest movement will affect the passions, and it plays the essential role of uniting the two symmetrical sides of the body. | |
| From: report of René Descartes (The Passions of the Soul [1649], §31) by PG - Db (ideas) | |
| A reaction: See Idea 4862 for Spinoza's nice response to Descartes' proposal. If Descartes had followed brain research for the last four hundred years, at what point would he have wavered? If every single part of the brain seems to 'interact', dualism looks unlikely. |
| 4015 | For Descartes passions are God-given preservers of the mind-body union [Descartes, by Taylor,C] |
| Full Idea: Descartes sees passions not as opinions, but as functional devices that the Creator has designed for us to help preserve the body-soul substantial union. | |
| From: report of René Descartes (The Passions of the Soul [1649]) by Charles Taylor - Sources of the Self §8 | |
| A reaction: I wonder what Descartes would have made of the theory of evolution? |
| 4313 | Are there a few primary passions (say, joy, sadness and desire)? [Descartes, by Cottingham] |
| Full Idea: Descartes says there are six primary passions (wonder, love, hatred, desire, joy and sadness); Spinoza says there are just three (joy, sadness and desire). | |
| From: report of René Descartes (The Passions of the Soul [1649]) by John Cottingham - The Rationalists p.172 | |
| A reaction: A dubious project. However, it is now agreed that there are a few (six?) basic universal facial expressions, to which these passions may correspond. |
| 23989 | There are six primitive passions: wonder, love, hatred, desire, joy and sadness [Descartes, by Goldie] |
| Full Idea: Descartes said there are six primitive passions, namely wonder, love, hatred, desire, joy and sadness. The others are either species of these, or composed of them. | |
| From: report of René Descartes (The Passions of the Soul [1649], 353) by Peter Goldie - The Emotions 4 'Evidence' | |
| A reaction: [not sure about ref] It's a nice touch to add 'wonder', which doesn't make it onto anyone else's list. |
| 23997 | Plato saw emotions and appetites as wild horses, in need of taming [Plato, by Goldie] |
| Full Idea: Plato had a conception of the emotions and our bodily appetites as being like wild horses, to be harnassed and controlled by reason. | |
| From: report of Plato (Phaedrus [c.366 BCE]) by Peter Goldie - The Emotions 4 'Education' | |
| A reaction: This seems to make Plato the patriarch of puritanism. See Symposium, as well as Phaedrus. But bringing up children can often seem like taming wild beasts. |
| 159 | Only a good philosopher can be a good speaker [Plato] |
| Full Idea: Unless a man becomes an adequate philosopher he will never be an adequate speaker on any subject. | |
| From: Plato (Phaedrus [c.366 BCE], 261a) | |
| A reaction: Depends. Hitler showed little sign of clear philosophical thinking, but the addition of lights and uniforms seemed to sweep reasonably intelligent people along with him. |
| 5946 | 'Phaedrus' pioneers the notion of philosophical rhetoric [Lawson-Tancred on Plato] |
| Full Idea: The purpose of the 'Phaedrus' is to pioneer the notion of philosophical rhetoric. | |
| From: comment on Plato (Phaedrus [c.366 BCE], Ch.10) by Hugh Lawson-Tancred - Plato's Republic and Greek Enlightenment | |
| A reaction: This is a wonderfully challenging view of what Plato was up to. One might connect it with Rorty's claim that philosophy should move away from epistemology and analysis, towards hermeneutics, which sounds to me like rhetoric. 'Phaedrus' is beautiful. |
| 158 | An excellent speech seems to imply a knowledge of the truth in the mind of the speaker [Plato] |
| Full Idea: If a speech is to be classed as excellent, does that not presuppose knowledge of the truth about the subject of the speech in the mind of the speaker. | |
| From: Plato (Phaedrus [c.366 BCE], 259e) | |
| A reaction: I like the thought that Plato's main interest was rhetoric, but with the view that the only good rhetoric is truth-speaking. It would be hard to admire a speech if you disagreed with it. |
| 20037 | Merely willing to walk leads to our walking [Descartes] |
| Full Idea: Our merely willing to walk has the consequence that our legs move and we walk. | |
| From: René Descartes (The Passions of the Soul [1649], 18), quoted by Rowland Stout - Action 1 'Volitionism' | |
| A reaction: Stout attributes this to Descartes' dualism, as if legs are separate from persons. Stout says the idea of a prior mental act is not usually now considered as part of an action, or even to exist at all. If the volition is intentional, there is a regress. |
| 155 | Beauty is the clearest and most lovely of the Forms [Plato] |
| Full Idea: Only beauty has the privilege of being the most clearly discerned and the most lovely of the forms. | |
| From: Plato (Phaedrus [c.366 BCE], 250e) | |
| A reaction: the motivation in Plato's theory |
| 143 | The two ruling human principles are the natural desire for pleasure, and an acquired love of virtue [Plato] |
| Full Idea: In each one of us there are two ruling and impelling principles: a desire for pleasure, which is innate, and an acquired conviction which causes us to aim at excellence. | |
| From: Plato (Phaedrus [c.366 BCE], 237d) | |
| A reaction: This division is too neat and simple. An obsession with pleasure I would take to be acquired. If you set out to do something, I think there is an innate desire to do it well. |
| 16763 | We don't die because the soul departs; the soul departs because the organs cease functioning [Descartes] |
| Full Idea: We ought to hold, on the contrary, that the soul takes its leave when we die only because this heat ceases and the organs that bring about bodily movement decay. | |
| From: René Descartes (The Passions of the Soul [1649], I.5), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 24.5 | |
| A reaction: This sounds like a pretty major change in our concept of death, given that we all now agree with Descartes. |
| 157 | Most pleasure is release from pain, and is therefore not worthwhile [Plato] |
| Full Idea: Life is not worth living for pleasures whose enjoyment entirely depends on previous sensation of pain, like almost all physical pleasures. | |
| From: Plato (Phaedrus [c.366 BCE], 258e) | |
| A reaction: Eating exotic food which is hard to obtain? (Pay someone to obtain it). Rock climbing. Training for sport. |
| 144 | Reason impels us towards excellence, which teaches us self-control [Plato] |
| Full Idea: The conviction which impels us towards excellence is rational, and the power by which it masters us we call self-control. | |
| From: Plato (Phaedrus [c.366 BCE], 237e) |
| 4016 | Descartes makes strength of will the central virtue [Descartes, by Taylor,C] |
| Full Idea: Descartes makes strength of will the central virtue. | |
| From: report of René Descartes (The Passions of the Soul [1649]) by Charles Taylor - Sources of the Self §8 | |
| A reaction: Presumably strength of will can serve evil ends, so this is a bit confusing. |
| 156 | Bad people are never really friends with one another [Plato] |
| Full Idea: It is not ordained that bad men should be friends with one another. | |
| From: Plato (Phaedrus [c.366 BCE], 255b) |
| 148 | If the prime origin is destroyed, it will not come into being again out of anything [Plato] |
| Full Idea: If the prime origin is destroyed, it will not come into being again out of anything. | |
| From: Plato (Phaedrus [c.366 BCE], 245d) | |
| A reaction: This is the essence of Aquinas's Third Way of proving God's existence. |
| 152 | The mind of God is fully satisfied and happy with a vision of reality and truth [Plato] |
| Full Idea: The mind of a god, sustained by pure intelligence and knowledge, is satisfied with the vision of reality, and nourished and made happy by the vision of truth. | |
| From: Plato (Phaedrus [c.366 BCE], 247d) |
| 150 | We cannot conceive of God, so we have to think of Him as an immortal version of ourselves [Plato] |
| Full Idea: Because we have never seen or formed an adequate idea of a god, we picture him to ourselves as a being of the same kind as ourselves but immortal. | |
| From: Plato (Phaedrus [c.366 BCE], 246d) |
| 149 | There isn't a single reason for positing the existence of immortal beings [Plato] |
| Full Idea: There is not a single sound reason for positing the existence of such a being who is immortal | |
| From: Plato (Phaedrus [c.366 BCE], 246d) |
| 146 | Soul is always in motion, so it must be self-moving and immortal [Plato] |
| Full Idea: All soul is immortal, for what is always in motion is immortal. Only that which moves itself never ceases to be in motion. | |
| From: Plato (Phaedrus [c.366 BCE], 245c) |